To solve for dy/dx and d²y/dx², we start by simplifying and differentiating the function step by step. The function given is:
y = 3sin(t) + 4cos(0t + 2π)
The term 4cos(0t + 2π) simplifies because cos(0t) is always 1, regardless of the value of t. Thus, we can rewrite the equation as:
y = 3sin(t) + 4
Now, we differentiate to find dy/dx:
1. The derivative of 3sin(t) is 3cos(t).
2. The derivative of 4 is 0 since it’s a constant.
Combining these results, we get:
dy/dx = 3cos(t)
Next, we find the second derivative, d²y/dx²:
1. The derivative of 3cos(t) is -3sin(t).
So, we have:
d²y/dx² = -3sin(t)
In summary, the calculations provide us with:
dy/dx = 3cos(t)
d²y/dx² = -3sin(t)
These derivatives indicate how the value of y changes with respect to t, and the second derivative gives insight into the acceleration of that change.